No. Technically, a divisional application requires only claims that were originally presented in the preceding case and not pursued.
For this reason, whenever you have several sequential divisionals stemming from a multi-way restriction requirement in the original parent case, and the benefit claim runs as a chain back through the sequentially filed divisionals for pendency, each divisional should be filed with a full set of claims from the parent, along with a preliminary amendment canceling the claims not being pursued in that particular child. For if you had filed the first divisional with only the set of claims being pursued in that case, the next divisional and thereafter technically are not divisionals of the directly preceding application (however, they could be considered continuations if the specification otherwise remained the same.)
For this reason, whenever you have several sequential divisionals stemming from a multi-way restriction requirement in the original parent case, and the benefit claim runs as a chain back through the sequentially filed divisionals for pendency, each divisional should be filed with a full set of claims from the parent, along with a preliminary amendment canceling the claims not being pursued in that particular child. For if you had filed the first divisional with only the set of claims being pursued in that case, the next divisional and thereafter technically are not divisionals of the directly preceding application (however, they could be considered continuations if the specification otherwise remained the same.)